## A low cost adaptive optics system using a membrane mirror

Optics Express, Vol. 6, Issue 9, pp. 175-185 (2000)

http://dx.doi.org/10.1364/OE.6.000175

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### Abstract

A low cost adaptive optics system constructed almost entirely of commercially available components is presented. The system uses a 37 actuator membrane mirror and operates at frame rates up to 800 Hz using a single processor. Numerical modelling of the membrane mirror is used to optimize parameters of the system. The dynamic performance of the system is investigated in detail using a diffractive wavefront generator based on a ferroelectric spatial light modulator. This is used to produce wavefronts with time-varying aberrations. The ability of the system to correct for Kolmogorov turbulence with different strengths and effective wind speeds is measured experimentally using the wavefront generator.

© Optical Society of America

## 1. Introduction

1. G. Vdovin and P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. **34**, 2968–2972 (1995). [CrossRef] [PubMed]

2. E. Steinhaus and S. G. Lipson, “Bimorph piezoelectric flexible mirror,” J. Opt. Soc. Am. **69**, 478–481 (1979). [CrossRef]

3. J. C. Dainty, A. V. Koryabin, and A. V. Kudryashov, “Low-order adaptive deformable mirror,” Appl. Opt. **37**, 4663–4668 (1998). [CrossRef]

6. S. Restaino, D. Dayton, S. Browne, J. Gonglewski, J. Baker, S. Rogers, S. McDermott, J. Gallegos, and M. Shilko, “On the use of dual frequency nematic material for adaptive optics systems: first results of a closed-loop experiment,” Opt. Express **6**, 2–6 (2000). http://www.opticsexpress.org/oearchive/source/18848.htm [CrossRef] [PubMed]

## 2. System description

*µ*m square pixels, which has a maximum frame rate of about 800Hz.

## 3. The membrane mirror

8. R. P. Grosso and M. Yellin, “The membrane mirror as an adaptive optical element,” J. Opt. Soc. Am. **67**, 399–406 (1977). [CrossRef]

8. R. P. Grosso and M. Yellin, “The membrane mirror as an adaptive optical element,” J. Opt. Soc. Am. **67**, 399–406 (1977). [CrossRef]

*z*is the deformation,

*T*the membrane tension and

*P*the electrostatic pressure due to the voltage

*V*across the gap

*d*between the membrane and the electrode, with the boundary condition

*z*=0 at the clamped edge of the membrane. A finite element model was used to solve the linear approximation (

*d*≈

*d*

_{0}) of this equation, to calculate the mirror influence functions.

_{m}, the effect of the resulting mirror deformation on the reflected wavefront in terms of a suitable orthonormal wavefront expansion is given by

**A**

_{m}is the influence matrix of the mirror. Conversely, the actuator signals which give the least-squares best fit to a required wavefront correction

*ϕ*

_{0}are given by

**A**and is given by

**U**,

**S**and

**V**are the singular value decomposition of

**A**

_{m}such that

**A**

_{m}=

**USV**

^{T}. The columns of the matrices

**U**and

**V**make up orthonormal sets of the mirror deformation and actuator signal spaces respectively and can be thought of as spatial modes of the system. The values of the diagonal matrix

**S**are the singular values and represent the ‘gains’ of the different modes: a small singular value implies that a large actuator signal is required to produce unit amplitude of the given deformation mode and vice versa. Thus, these values, and in particular the ratio of the smallest to the largest value (the condition factor), give a measure of the controllability of the mirror. Modes with small singular values can have a disproportionate effect on the control matrix, resulting in large actuator signals and hence actuator clipping, and are sensitive to wavefront sensor noise. Discarding such modes may improve the controllability of the system. Fig 3 shows the singular values for the influence matrix of the TU Delft membrane mirror using different optical pupil diameters. The range of singular values is large (two and three orders of magnitude for

*D*=

*D*

_{m}and

*D*=0.5

*D*

_{m}respectively). Although it is necessary to choose an optical pupil smaller than the membrane diameter to allow for non-zero phase at the edge of the pupil [10

10. E. S. Claflin and N. Bareket, “Configuring an electrostatic membrane mirror by least- squares fitting with analytically derived influence functions,” J. Opt. Soc. Am. A **3**, 1833–1839 (1986). [CrossRef]

*D*=

*D*

_{m}.

### 3.1. Accounting for the aberration statistics

*k*|

^{-11/3}. This will give a weighting to the expected magnitudes of the various mirror modes required for aberration compensation. To get a better idea of the controllability of the membrane for specific types of aberration, it is useful to include these statistics in the analysis. If the Kahunen-Loève [11] expansion is chosen for the orthonormal wavefront expansion, then the covariance of

*ϕ*denoted

**C**=〈

*ϕϕ*

^{T}〉 is diagonal. Furthermore, multiplying the wavefront expansion by C

^{-1/2}gives a vector C

^{-1/2}

*ϕ*, which has a covariance equal to the identity matrix. Taking these weights for the different modes of the expansion into account in the influence matrix, Eq. (2) describing the mirror deformations becomes

^{-1/2}

**A**

_{m}gives a better measure of the controllability of the mirror for wavefront aberrations of these statistics. Fig 5 shows the singular values for the influence matrix weighted for Kolmogorov statistics. The ranges of singular values are reduced compared with those for the unweighted matrix, particularly for the larger optical pupil sizes, suggesting that the spatial response of the mirror is suited to correction of Kolmogorov aberrations.

*k*to vary with

*k*

^{-2}. This is in fact very similar to the Kolmogorov power law where the typical amplitude of the aberrations varies with

*k*

^{-11/6}.

### 3.2. Numerical Modelling

*ϕ*

_{0}, using least-squares correction, and clipping the actuator signals to their maximum permissible values with the limiting function

*L*(x), the residual wavefront error (fitting error) is given by

*L*(x) is given by

*x*

_{max}is the maximum actuator signal permissible. The piston term, which has no effect on the image, was ignored throughout by excluding it from the basis used to expand the wavefront. Then, the residual wavefront variance over the aperture is given by

*D*/

*r*

_{0}=7 and

*D*/

*r*

_{0}=10 with the TU Delft mirror are shown in Fig 6. Figures 6(a) and (b) show the effect of the choice of the optical pupil diameter and the number spatial correction modes (discarding those modes with the smallest singular values) on the mean Strehl, when using the mirror to correct aberrations with Kolmogorov statistics for two strengths. The approximation

*S*=exp(-

*D*=0.65

*D*

_{m}. Figures 6(c) and (d) show respectively the effects of the choice of optical pupil size and of the number of spatial control modes in the control matrix, discarding those with the smallest singular values. Discarding modes at the higher aberration strengths increases the Strehl achievable. Note that this is not a result of wavefront sensor noise, which this model does not account for, but is a result of actuator clipping which is more pronounced at the higher aberration strengths. The effect of wavefront sensor noise will be greatest for spatial modes with the smallest singular values (since they result in modes with large gains in the control matrix) and is likely to degrade performance with large numbers of modes further. The results show, however, that it should be possible to achieve reasonable correction (Strehl ratio of 0.7) for Kolmogorov turbulence with

*D*/

*r*

_{0}=7 using the 37 actuator membrane mirror.

## 4. System Operation

*n*th iteration of the control loop are given by

*g*is the integrator gain,

**M**is the spatial control matrix, s is the sensor signal and

*β*(

*β*≪1)is a bleed parameter included to make the system resistant to possible modes of the mirror which are invisible to the wavefront sensor. In fact we have not seen any evidence of such modes in the system, which is probably attributable to the unmatched nature of the mirror-sensor system. A curvature wavefront sensor would be better matched to the membrane mirror. Suitable values for the temporal control loop parameters were found by trial end error.

## 5. System Performance

12. M. A. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. **23**, 1849–1851 (1998). [CrossRef]

^{1}The video (Fig 7) shows a sample of the dynamic behaviour of the system when correcting for these wavefronts.

*D*/

*r*

_{0}and

*v*/

*r*

_{0}, where

*r*

_{0}is the Fried parameter,

*D*the input aperture diameter and

*v*the speed which the frozen turbulence layer moves across the input aperture, i.e., the wind speed. Long exposures recorded with the science camera were used to calculated the Strehl ratio of the corrected and uncorrected output images.

*v*/

*r*

_{0}), for two different strengths (

*D*/

*r*

_{0}). For small

*v*/

*r*

_{0}, the Strehl ratio achieved is in agreement with that predicted for the mirror by the static model [Fig 6(d)]. Also note that considerable improvement to the Strehl ratio is achieved even with normalized wind speeds

*v*/

*r*

_{0}in excess of 100 Hz

## 6. Conclusions

*v*/

*r*

_{0}) in excess of 100 Hz. It has been demonstrated experimentally that the performance of the system can be improved by discarding a number of spatial modes from the system control matrix. There are still a number of improvements which could be made to the system’s spatial and temporal control. In particular, the temporal control of the system is far from optimal and it is expected that optimization of this this should give considerable improvements to the closed-loop bandwidth of the system.

## Acknowledgements

## Footnotes

1 | In fact the turbulent screens generated are not pure Kolmogorov in nature. Since they are generated as the inverse discrete Fourier transform of sampled Kolmogorov spectra [13 13. A. Glindemann, R. G. Lane, and J. C. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. |

## References and links

1. | G. Vdovin and P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. |

2. | E. Steinhaus and S. G. Lipson, “Bimorph piezoelectric flexible mirror,” J. Opt. Soc. Am. |

3. | J. C. Dainty, A. V. Koryabin, and A. V. Kudryashov, “Low-order adaptive deformable mirror,” Appl. Opt. |

4. | D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, and V. Biliotti, “Adaptive optics wave-front corrector using addressable liquid-crystal retarders.2.,” In |

5. | G. D. Love, “Wave-front correction and production of Zernike modes with a liquid- crystal spatial light modulator,” Appl. Opt. |

6. | S. Restaino, D. Dayton, S. Browne, J. Gonglewski, J. Baker, S. Rogers, S. McDermott, J. Gallegos, and M. Shilko, “On the use of dual frequency nematic material for adaptive optics systems: first results of a closed-loop experiment,” Opt. Express |

7. | |

8. | R. P. Grosso and M. Yellin, “The membrane mirror as an adaptive optical element,” J. Opt. Soc. Am. |

9. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

10. | E. S. Claflin and N. Bareket, “Configuring an electrostatic membrane mirror by least- squares fitting with analytically derived influence functions,” J. Opt. Soc. Am. A |

11. | F. Roddier, “The problematic of adaptive optics design,” in |

12. | M. A. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. |

13. | A. Glindemann, R. G. Lane, and J. C. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 4, 2000

Published: April 24, 2000

**Citation**

Carl Paterson, I. Munro, and J. Dainty, "A low cost adaptive optics system using a membrane mirror," Opt. Express **6**, 175-185 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-9-175

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### References

- G. Vdovin and P. M. Sarro, "Flexible mirror micromachined in silicon," Appl. Opt. 34, 2968-2972 (1995). [CrossRef] [PubMed]
- E. Steinhaus and S. G. Lipson, "Bimorph piezoelectric flexible mirror," J. Opt. Soc. Am. 69, 478-481 (1979). [CrossRef]
- J. C. Daint , A. V. Koryabin, and A. V. Kudryashov, "Low-order adaptive deformable mirror," Appl. Opt. 37, 4663-4668 (1998). [CrossRef]
- D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, and V. Biliotti, "Adaptive optics wave-front corrector using addressable liquid-crystal retarders .2.," In Active and adaptive optical components, Proc. SPIE 1543, 133-143 (Osserv Astrofis Arcetri, I-50125 Florence, Ital , 1992).
- G. D. Love, "Wave-front correction and production of Zernike modes with a liquid- crystal spatial light modulator," Appl. Opt. 36, 1517-1524 (1997). [CrossRef] [PubMed]
- S. Restaino, D. Dayton, S. Browne, J. Gonglewski, J. Baker, S. Rogers, S. McDermott, J. Gallegos, and M. Shilko, "On the use of dual frequenc nematic material for adaptive optics systems: first results of a closed-loop experiment," Opt. Express 6, 2-6 (2000). http://www.opticsexpress.org/oearchive/source/18848.htm [CrossRef] [PubMed]
- http://okotech.com/mirrors/technical/index.html
- R. P. Grosso and M. Yellin, "The membrane mirror as an adaptive optical element," J. Opt. Soc. Am. 67, 399-406 (1977). [CrossRef]
- W. H. Press, S. A. Teukolsk , W. T. Vetterling, and B. P. Flanner, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992).
- E. S. Claflin and N. Bareket, "Configuring an electrostatic membrane mirror by least- squares fitting with analytically derived influence functions," J. Opt. Soc. Am. A 3, 1833-1839 (1986). [CrossRef]
- F. Roddier, "The problematic of adaptive optics design," in Adaptive optics for astronomy, D. M. Alloin and J. M. Mariotti, eds., (Kluwer Academic, 1994), pp. 89-111.
- M. A. A. Neil, M. J. Booth, and T. Wilson, "Dynamic wave-front generation for the characterization and testing of optical systems," Opt. Lett. 23, 1849-1851 (1998). [CrossRef]
- A. lindemann, R. G. Lane, and J. C. Daint, "Simulation of time-evolving speckle patterns using Kolmogorov statistics," J. Mod. Opt. 40, 2381-2388 (1993). [CrossRef]

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